A145602 -id:A145602 - cp's OEIS frontend

A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500

Offset: 0

N. J. A. Sloane

nonn

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003
Number of tilings of an  hexagon.
a(n) is the number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan, Jul 25 2005
The number of returning walks of length 2n on the upper half of a square lattice, since a(n) = Sum_{k=0..2n} binomial(2n,k)*A126120(k)*A126869(n-k). - Andrew V. Sutherland, Mar 24 2008
For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). - Peter Bala, Oct 14 2008
The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012
a(n) is number of pairs of non-intersecting lattice paths from (0,0) to (n+1,n+1) using (1,0) and (0,1) as steps. Here, non-intersecting means two paths do not share a vertex except the origin and the destination. For example, a(1) = 3 because we have three such pairs from (0,0) to (2,2): {NNEE,EENN}, {NNEE,ENEN}, {NENE,EENN}. - Ran Pan, Oct 01 2015
Also the number of ordered rooted trees with 2(n+1) nodes and n+1 leaves, i.e., half of the nodes are leaves. These trees are ranked by A358579. The unordered version is A185650. - Gus Wiseman, Nov 27 2022
The number of secondary GL(2) invariants constructed from n+1 two component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025

			G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
From _Gus Wiseman_, Nov 27 2022: (Start)
The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
  (((o)oo))  (((o)o)o)  (((o))oo)
  (((oo)o))  (((oo))o)  ((o)(o)o)
  (((ooo)))  ((o)(oo))  ((o)o(o))
  ((o(o)o))  ((o(o))o)  (o((o))o)
  ((o(oo)))  ((oo)(o))  (o(o)(o))
  ((oo(o)))  (o((o)o))  (oo((o)))
             (o((oo)))
             (o(o(o)))
(End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Elena Barcucci, Andrea Frosini and Simone Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838.
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Ivan Marin and Emmanuel Wagner, A cubic defining algebra for the Links-Gould polynomial. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From _N. J. A. Sloane_, Sep 21 2012
Henri Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Guoce Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A000356, A010370, A038535.
Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008
Cf. A000142, A000894, A248045.
Equals half of A267981.
Counts the trees ranked by A358579.
A001263 counts ordered rooted trees by nodes and leaves.
A090181 counts ordered rooted trees by nodes and internals.
Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.

a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013

[Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

with(combstruct): bin := {B=Union(Z,Prod(B,B))} :seq(1/2*binomial(2*i,i)*(count([B,bin,unlabeled],size=i)), i=1..18) ; # Zerinvary Lajos, Jun 06 2007

a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ BesselI[0, 2 x] BesselI[1, 2 x] / x, {x, 0, 2 n}]]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* Michael Somos, Sep 18 2016 *)
a[n_] := (2 n + 1) CatalanNumber[n]^2;
Array[a, 20, 0] (* Peter Luschny, Mar 03 2020 *)

{a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */

a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ Hugo Pfoertner, Oct 22 2022

-4*a(n) = A010370(n+1).
G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. [edited by Olivier Gérard, Feb 16 2011]
G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x) * BesselI(1, 2*x) / x. - Michael Somos, Jun 22 2005
a(n) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007
For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - Gary W. Adamson, Apr 08 2011
a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle).
G.f.: If G_N(x) = 1 + Sum_{k=1..N} ((2*k)!*(2*k+1)!*x^k)/(k!*(k+1)!)^2, G_N(x) = 1 + 12*x/(G(0) - 12*x); G(k) = 16*x*k^2 + 32*x*k + k^2 + 4*k + 12*x + 4 - 4*x*(2*k+3)*(2*k+5)*(k+2)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
D-finite with recurrence (n+1)^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014
a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: 2F2(1/2,3/2; 2,2; 16*x).
a(n) ~ 2^(4*n+1)/(Pi*n^2). (End)
a(n) = A005408(n)*(A000108(n))^2. - Ivan N. Ianakiev, Nov 13 2019
a(n) = det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(n+j+1,i+1). - Benoit Cloitre, Oct 22 2022
a(n) = Integral_{x=0..16} x^n*W(x) dx, where W(x) = (16*EllipticE(1 - x/16) - x*EllipticK(1 - x/16))/(8*Pi^2*sqrt(x)), n=>0. W(x) diverges at x=0, monotonically decreases for x>0, and vanishes at x=16. EllipticE and EllipticK are elliptic functions. This integral representation as n-th moment of a positive function W(x) on the interval [0, 16] is unique. - Karol A. Penson, Dec 20 2023

More terms from Andrew V. Sutherland, Mar 24 2008

A145600 a(n) is the number of walks from (0,0) to (0,1) that remain in the upper half-plane y >= 0 using (2*n - 1) unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700

Offset: 1

Peter Bala, Oct 14 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145601, A145602 and A145603. This sequence is the central column taken from triangle A145596, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 1.

			a(2) = 8: the 8 walks from (0,0) to (0,1) of three steps are
UDU, UUD, URL, ULR, RLU, LRU, RUL and LUR.

M. Dukes and Y. Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013, Pages 816-842. - From N. J. A. Sloane, Feb 21 2013

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Cf. A000891, A145596, A145601, A145602, A145603.

a(n) := 1/n*binomial(2*n,n+1)*binomial(2*n,n-1);
seq(a(n),n = 1..19);

a(n) = 1/n*binomial(2*n,n+1)*binomial(2*n,n-1).
a(n) = A135389(n-1)/(n+1). - R. J. Mathar, Jul 14 2013
D-finite with recurrence (n+1)^2*a(n) -4*n*(5*n-1)*a(n-1) +16*(2*n-3)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013

A145601 a(n) is the number of walks from (0,0) to (0,2) that remain in the upper half-plane y >= 0 using 2*n unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 15, 189, 2352, 29700, 382239, 5010005, 66745536, 901995588, 12342120700, 170724392916, 2384209771200, 33577620944400, 476432168185575, 6805332732133125, 97790670976838400, 1412830549632694500

Offset: 1

Peter Bala, Oct 15 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145602 and A145603. This sequence is the central column taken from triangle A145597, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 2.

			a(2) = 15: the 15 walks from (0,0) to (0,2) of four steps are:
UUUD, UULR, UURL, UUDU, URUL, ULUR, URLU, ULRU,RUUL, LUUR,
RLUU, LRUU, RULU, LURU and UDUU.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

A000891, A145597, A145600, A145602, A145603.

with(combinat):
a(n) = 3/(2*n+1)*binomial(2*n+1,n+2)*binomial(2*n+1,n-1);
seq(a(n),n = 1..19);

a(n) = 3/(2*n+1)*binomial(2*n+1,n+2)*binomial(2*n+1,n-1).

A145603 a(n) is the number of walks from (0,0) to (0,4) that remain in the upper half-plane y >= 0 using 2*n +2 unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 35, 720, 12375, 196625, 3006003, 45048640, 668144880, 9859090500, 145173803500, 2136958387520, 31479019635375, 464342770607625, 6861343701121875, 101583106970400000, 1507019252941540800

Offset: 1

Peter Bala, Oct 15 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145601 and A145602. This sequence is the central column taken from the triangle A145599, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 4.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

A000891, A145599, A145600, A145601, A145602.

with(combinat):
a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1);
seq(a(n),n = 1..19);

a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1).

G.f.: x*hypergeom([5/2, 3, 3, 7/2], [1, 6, 6], 16*x). - Stefano Spezia, Aug 26 2025

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4binomial(n+1, k+3)binomial(n+1, k-1)/(n+1).

Original entry on oeis.org

1, 4, 4, 10, 24, 10, 20, 84, 84, 20, 35, 224, 392, 224, 35, 56, 504, 1344, 1344, 504, 56, 84, 1008, 3780, 5760, 3780, 1008, 84, 120, 1848, 9240, 19800, 19800, 9240, 1848, 120, 165, 3168, 20328, 58080, 81675, 58080, 20328, 3168, 165, 220, 5148, 41184, 151008

Offset: 3

Peter Bala, Oct 15 2008

T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 3 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.

The current array is the case r = 3 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145599 (r = 4).

			Triangle starts
  n\k|  1     2     3     4     5     6
  =====================================
   3 |  1
   4 |  4     4
   5 | 10    24    10
   6 | 20    84    84    20
   7 | 35   224   392   224    35
   8 | 56   504  1344  1344   504    56
  ...
Row 5: T(5,3) = 10: the 10 walks of length 5 from (0,0) to (2,3) are UUURR, UURUR, UURRU, URUUR, URURU, URRUU, RUUUR, RUURU, RURUU and RRUUU.
*
*......*......*......y......*......*......*
.
.
*.....10......*.....24......*.....10......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......o......*......*......* x axis
.

F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=3.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Cf. A003518 (row sums), A001263, A145602, A145596, A145597, A145599.

T := (n,k) -> 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1):
for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do;

T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1) for n >=3 and 1 <= k <= n-2. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 1,3). Row sums A003518.
O.g.f. for column k+2: 4/(k + 1) * y^(k+4)/(1 - y)^(k+6) * Jacobi_P(k,4,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) = Sum_{k = 1 .. n - 2} T(n,k)*x^k:
x^3*R_(n-1)(x) = 4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)*(n + 4)) * Sum_{k = 0..n} binomial(n + 4,k) * binomial(2n - k,n) * (x - 1)^k;
Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-1) = A003518(n)*x^(n-1).
Row generating polynomial R_(n+3)(x) = 4/(n+4)*x*(1-x)^n * Jacobi_P(n,4,4,(1+x)/(1-x)). - Peter Bala, Oct 31 2008
G.f.: A(x) = x*A145596(x)^2. - Vladimir Kruchinin, Oct 09 2020

A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 20, 0, 15, 0, 1, 0, 75, 0, 24, 0, 1, 175, 0, 189, 0, 35, 0, 1, 0, 784, 0, 392, 0, 48, 0, 1, 1764, 0, 2352, 0, 720, 0, 63, 0, 1, 0, 8820, 0, 5760, 0, 1215, 0, 80, 0, 1, 19404, 0, 29700, 0, 12375, 0, 1925, 0, 99, 0, 1

Offset: 0

Peter Luschny, Dec 07 2024

Consider square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. T(n, k) is the number of walks with length n and height k. The number of walks with positive height is A378060, and with nonnegative height is A018224. Walks of odd length can never have an even height, and walks of even length cannot have an odd height. The Python program below generates the walks.

			Triangle starts:
  0  [   1]
  1  [   0,    1]
  2  [   3,    0,    1]
  3  [   0,    8,    0,    1]
  4  [  20,    0,   15,    0,   1]
  5  [   0,   75,    0,   24,   0,    1]
  6  [ 175,    0,  189,    0,  35,    0,  1]
  7  [   0,  784,    0,  392,   0,   48,  0,  1]
  8  [1764,    0, 2352,    0, 720,    0, 63,  0, 1]
  9  [   0, 8820,    0, 5760,   0, 1215,  0, 80, 0, 1]
.
The 15 walks with length 4 and height 2 are: 'NNNS', 'NNSN', 'NNWE', 'NNEW', 'NSNN', 'NWNE', 'NWEN', 'NENW', 'NEWN', 'WNNE', 'WNEN', 'WENN', 'ENNW', 'ENWN', 'EWNN'.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

The columns are aerated rows of A378062. See also: A000891, A145600, A145601, A145602, A145603.

Cf. A018224 (row sums), A378060.

T := (n, k) -> ifelse((n - k)::odd, 0, binomial(n+1, (n-k)/2)^2*(k+1)/(n+1)):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

T[n_, k_] := If[EvenQ[n-k],Binomial[n + 1, (n - k)/2]^2*(k + 1)/(n + 1), 0]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

# Creates the table by counting the heights of square lattice walks. For illustration only.
from dataclasses import dataclass
@dataclass
class Z: w: str = ""; r: int = 0; i: int = 0
def Trow(n: int) -> list[int]:
    W = [Z()]
    row = [0] * (n + 1)
    for x in W:
        if len(x.w) == n:
            if x.r == 0: row[x.i] += 1
        else:
            for s in "NSWE":
                r = i = 0
                match s:
                    case "W": r = 1
                    case "E": r = -1
                    case "N": i = 1
                    case "S": i = -1
                if x.i + i >= 0:
                    W.append(Z(x.w + s, x.r + r, x.i + i))
    return row
for n in range(10): print(f"[{n}] {Trow(n)}")

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 8, 20, 0, 1, 15, 75, 175, 0, 1, 24, 189, 784, 1764, 0, 1, 35, 392, 2352, 8820, 19404, 0, 1, 48, 720, 5760, 29700, 104544, 226512, 0, 1, 63, 1215, 12375, 81675, 382239, 1288287, 2760615, 0, 1, 80, 1925, 24200, 196625, 1145144, 5010005, 16359200, 34763300

Offset: 0

Peter Luschny, Dec 07 2024

			Array A(n, k) starts:
  [0] 0, 1,  3,   20,   175,    1764,    19404, ... A000891
  [1] 0, 1,  8,   75,   784,    8820,   104544, ... A145600
  [2] 0, 1, 15,  189,  2352,   29700,   382239, ... A145601
  [3] 0, 1, 24,  392,  5760,   81675,  1145144, ... A145602
  [4] 0, 1, 35,  720, 12375,  196625,  3006003, ... A145603
  [5] 0, 1, 48, 1215, 24200,  429429,  7154784, ...
  [6] 0, 1, 63, 1925, 44044,  869505, 15767024, ...
  [7] 0, 1, 80, 2904, 75712, 1656200, 32626944, ...
.
Seen as a triangle, T(n, k) = A(n-k, k). Compare the descending antidiagonals of A378061.
  [0] 0;
  [1] 0, 1;
  [2] 0, 1,  3;
  [3] 0, 1,  8,  20;
  [4] 0, 1, 15,  75,  175;
  [5] 0, 1, 24, 189,  784,  1764;
  [6] 0, 1, 35, 392, 2352,  8820,  19404;
  [7] 0, 1, 48, 720, 5760, 29700, 104544, 226512;

Rows: A000891, A145600, A145601, A145602, A145603.

Columns: A005563, A005565, A378061.

A := (n, k) -> ifelse(k = 0, 0, (n + 1)*binomial(2*k + n - 1, k - 1)^2/(2*k + n - 1)):
for n from 0 to 7 do seq(A(n, k), k = 0..7);

A[n_, k_] := If[k==0, 0, (n + 1)*Binomial[2*k + n - 1, k - 1]^2 / (2*k + n - 1)]; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

cp's OEIS Frontend

A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2.

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A145600 a(n) is the number of walks from (0,0) to (0,1) that remain in the upper half-plane y >= 0 using (2*n - 1) unit steps either up (U), down (D), left (L) or right (R).

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Maple

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A145601 a(n) is the number of walks from (0,0) to (0,2) that remain in the upper half-plane y >= 0 using 2*n unit steps either up (U), down (D), left (L) or right (R).

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Maple

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A145603 a(n) is the number of walks from (0,0) to (0,4) that remain in the upper half-plane y >= 0 using 2*n +2 unit steps either up (U), down (D), left (L) or right (R).

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A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4*binomial(n+1, k+3)*binomial(n+1, k-1)/(n+1).

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A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

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A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)*binomial(2*k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

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A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4binomial(n+1, k+3)binomial(n+1, k-1)/(n+1).

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.